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E. Almqvist 3 years ago
parent 8522316370
commit 08ee4d429a
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  1. 20
      ma5/notes/diffequ/2022-04-05.aux
  2. 84
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  3. 553
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  4. 400
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  6. BIN
      ma5/notes/diffequ/2022-04-05.pdf
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\documentclass{article}
\usepackage[margin=2cm]{geometry}
\usepackage{titlesec}
\usepackage{titling}
\usepackage[hidelinks]{hyperref}
\usepackage{multicol}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{braket}
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{}
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{}[\titlerule]
\titleformat{\subsection}
{\large\bfseries}
{}
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\renewcommand{\maketitle}{
\begin{center}
{\huge\bfseries\thetitle}\\
\vspace{1em}
{\Large\theauthor} \\
\vspace{1em}
elalmqvist@gmail.com --- \url{https://wych.dev}
\end{center}
}
\begin{document}
\title{Anteckningar 2022-04-05}
\author{Elias Almqvist}
\maketitle
\newpage
\section{Inhomogena diff-ekvationer av första-ordningen bevis}
$$
y' + ay = f(x)
$$
$$
\text{Låt } g = y_{p1} - y_{p2}
$$
Där $y_{p1}$ och $y_{p2}$ (antagande) är partikulärlösningar till ekvationen $y' + ay = f(x)$. Alltså är $y_{pn}' + ay_{pn} = f(x)$ Detta ger den homogena ekvationen för $g$:
$$
g' + ag = y_{p1}' - y_{p2}' + a\left(y_{p1} - y_{p2}\right) = \left( y_{p1}' + ay_{p1} \right) - \left( y_{p2}' + ay_{p2} \right)
$$
Eftersom $\left( y_{p1}' + ay_{p1} \right) - \left( y_{p2}' + ay_{p2} \right) = f(x) - f(x)$ får vi att:
$$
g' + ag = \left( y_{p1}' + ay_{p1} \right) - \left( y_{p2}' + ay_{p2} \right) = f(x) - f(x) = 0
$$
Dvs $g = y_{p1} - y_{p2}$ är en lösning till $y' + ay = 0$
$$
\implies g + y_{p2} = y_{p1}
$$
$$
\therefore y = g + y_{p2} \because g = y_{p1} - y_{p2}
$$
Då $g = y_h$ och $y_{p2} = y_{p}$
$$
y = y_h + y_p \quad \text{V.S.V}
$$
\end{document}
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